580 GEOMETRY 



tion of the conditions for the protective equivalence of surfaces in 

 terms of their fundamental quadratic forms. 



Coordinate with what has just been stated, that general configura- 

 tions may be studied from the protective point of view, is the fact 

 that algebraic configurations may be studied in relation to general 

 transformation theory. One may object that, with respect to the 

 group of all (analytic) point transformations, the algebraic con- 

 figurations do not form a body, 1 that is, are not converted into 

 algebraic configurations; but such a body is obtained by adjoining 

 to the algebraic all those transcendental configurations which are 

 equivalent to algebraic. As this appears to have been overlooked, 

 it seems desirable to give a few concrete instances, of interest in 

 showing the effect of looking at familiar objects from a new and 

 more general point of view. 



As a first example, consider the idea of a linear system of plane 

 curves. In algebraic geometry, a linear system is understood to be 

 one represented by an equation of the form 





where the X's are parameters and the F's are polynomials hi x,y. On 

 the other hand, in general (infinitesimal) geometry, a system is defined 

 to be linear when it can be reduced (by the introduction of new 

 parameters) to the same form where the F's are arbitrary functions. 

 The first definition is invariant under the projective group; the sec- 

 ond, under the group of all point transformations. If now we apply 

 the second definition to algebraic curves, the result does not coincide 

 with that given by the first definition. Thus, every one-parameter 

 system is linear in the general sense, while only pencils of curves are 

 linear in the projective sense. The first case of real importance is, 

 however, the two-parameter system, since here each point of view 

 gives restricted, though not identical, types. An example in point 

 is furnished by the vertical parabolas tangent to a fixed line, the 

 equation of the system being y = (ax+b) 2 . From the algebraic or 

 projective point of view, this is a quadratic system since the para- 

 meters are involved to the second degree; but the system is linear 

 from the general point of view since its equation may be written 

 ax+b V^=0. This suggests the problem: Determine the systems 

 of algebraic curves which are linear in the general sense. 



As a second example, consider, from both points of view, the 

 equivalence of pencils of straight lines in the plane. By means of 

 collineations any two pencils may be converted into any other two; 



1 The most extensive group for which the algebraic configurations form a body 

 consists of all algebraic transformations. It is rather remarkable that even this 

 theory has received no development. 



1 Halphen, Laguerre, Forsyth. This theory has been extended to simultaneous 

 equations and applied geometrically by E. J. Wilczynski (Trans. Amer. Math. 

 Soc., 1901-1904). 



